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While free fermion topological crystalline insulators have been largelyclassified, the analogous problem in the strongly interacting case has beenonly partially solved. In this paper, we develop a characterization andclassification of interacting, invertible fermionic topological phases in (2+1)dimensions with charge conservation, discrete magnetic translation and $M$-foldpoint group rotation symmetries, which form the group $G_f = \text{U}(1)^f\times_{\phi} [\mathbb{Z}^2\rtimes \mathbb{Z}_M]$ for $M=1,2,3,4,6$. $\phi$ isthe magnetic flux per unit cell. We derive a topological response theory interms of background crystalline gauge fields, which gives a completeclassification of different phases and a physical characterization in terms ofquantized response to symmetry defects. We then derive the same classificationin terms of a set of real space invariants $\{\Theta_{\text{o}}^\pm\}$ that canbe obtained from ground state expectation values of suitable partial rotationoperators. We explicitly relate these real space invariants to the quantizedcoefficients in the topological response theory, and find the dependence of theinvariants on the chiral central charge $c_-$ of the invertible phase. Finally,when $\phi = 0$ we derive an explicit map between the free and interactingclassifications.

Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups